barcode print in asp net have written ln (u2 + 6) instead of ln u2 + 6 because u2 + 6 > 0. in VS .NET

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have written ln (u2 + 6) instead of ln u2 + 6 because u2 + 6 > 0.
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INVERSE TRIGONOMETRIC FUNCTIONS
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[CHAP. 37
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37.11 Show that sin 1 x and tan 1 x are odd functions.
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More generally, we can prove that if a one-one function is odd (as are the restricted sin x and the restricted tan x, but not the restricted cot x), then its inverse function is odd. In fact, if f is an odd one-one function and g is its inverse, then g( f (x)) = g(f ( x)) = x = g(f (x)) [since f (x) = f ( x)] [since g is inverse of f ] [since g is inverse of f ]
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which shows that g is odd. (Could a one-one function be even )
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37.12 For each of the following functions f , determine whether it is one-one; and if it is, nd a formula for the inverse function f 1 . 3x 5 x (b) f (x) = |x| (c) f (x) = (a) f (x) = + 3 2 x+2 1 (e) f (x) = (x 1)5 (f ) f (x) = (d) f (x) = (x 1)4 x 1 x+2 3x + 5 (i) f (x) = (g) f (x) = (h) f (x) = 3 x x 1 x +4 37.13 Evaluate: (a) cos 1 (e) sec 1 2 2 2 2 3 (f ) sec 1 3 (c) tan 1 1 (g) csc 1 ( 2) (d) tan 1
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(b) sin 1
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(h) cot 1 ( 1)
(i) sec 1 ( 2) 37.14 (a) Let = cos 1 ( 1 ). Find the values of sin , tan , cot , sec , and csc . 3 (b) Let = sin 1 ( 1 ). Find cos , tan , cot , sec , and csc . 4 37.15 Compute the following values: 4 (a) sin cos 1 5 1 1 tan 1 2 (d) sin cos 5
(b) tan sec 1
13 5
(c) cos sin 1
3 + sec 1 3 5
(e) sin 1 (sin )
[Hints: Part (b) 52 + 122 = 132 ; part (c) cos (u + v) = cos u cos v sin u sin v; part (e) is a trick question.] 37.16 Find the domain and the range of the function f (x) = cos (tan 1 x). 37.17 Differentiate: (a) sin 1 x + cos 1 x (b) tan 1 x + cot 1 x (c) sec 1 x + csc 1 x (d) tan 1 x + tan 1 1 x
(e) Explain the signi cance of your answers.
CHAP. 37]
INVERSE TRIGONOMETRIC FUNCTIONS
37.18 Differentiate: (a) x tan 1 x (e) ex cos 1 x a+x (i) tan 1 1 ax (b) sin 1
(f ) ln (tan 1 x) 1 (j) sin 1 + sec 1 x x
(c) tan 1 (cos x) 1 (g) csc 1 x 1 2 (k) tan x
(d) ln (cot 1 3x) (h) x a2 x 2 + a2 sin 1 x a
37.19 What identity is implied by the result of Problem 37.18(i)
37.20 Find the following antiderivatives: dx (a) (b) 4 + x2 dx (f ) (e) (x 3) x 2 6x + 8 dx (i) (j) x 9x 2 16 (m) (q) dx 6x x 2 x dx 4 x4 (n) (r)
dx 4 + 9x 2 dx 3 2x 2 dx x 3x 2 2 2x dx 6x x 2 ex dx 4 + e2x
(c) (g) (k) (o) (s)
dx 25 x 2 dx 2 + 7x 2 dx (1 + x) x x dx x 2 + 8x + 20 cos x dx 5 + sin2 x
(d) (h) (l) (p) (t)
dx 25 16x 2 dx x x2 4 x dx x4 + 9 x 3 dx x 2 2x + 4 dx x 2 + 2x + 10
[Hints: (b) let u = 3x; (d) let u = 4x; (e) let u = x 3; (l) let u2 = x 4 + 9; (m) complete the square in x 2 6x; (n) Dx (6x x 2 ) = 6 2x; (p) divide x 3 by x 2 2x + 4.] 37.21 Find an equation of the tangent line to the graph of y = sin 1 (x/3) at the origin.
37.22 A ladder which is 13 feet long leans against a wall. The bottom of the ladder is sliding away from the base of the wall at the rate of 5 feet per second. How fast is the radian measure of the angle between the ladder and the ground changing at the moment when the bottom of the ladder is 12 feet from the base of the wall
37.23 The beam from a lighthouse 3 miles from a straight coastline turns at the rate of 5 revolutions per minute. How fast is the point P at which the beam hits the shore moving when that point is 4 miles from the point A on the shore directly opposite the lighthouse (see Fig. 37-9)
Fig. 37-9
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